## Welcome to BrainDen.com - Brain Teasers Forum

Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-) |

# Triangle, trisected

### #1

Posted 11 March 2013 - 09:08 AM

**Warm-up problem:**

Bisect the angles of a triangle.

Describe the point(s) where each bisector first intersects one of the others.

**Now try this:**

Trisect the angles of a triangle.

Describe the points where each trisector first intersects one of the others.

*Vidi vici veni.*

### #2

Posted 11 March 2013 - 12:38 PM

Do we have to name that point?

### #3

Posted 11 March 2013 - 03:19 PM

Do we have to name that point?

Spoiler for

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle **first** intersects a trisector of one of the other angles.

And there is something special about those three points.

*Vidi vici veni.*

### #4

Posted 12 March 2013 - 09:49 AM

Do we have to name that point?

Spoiler for

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle

firstintersects a trisector of one of the other angles.And there is something special about those three points.

### #5

Posted 14 March 2013 - 04:38 PM

Do we have to name that point?

Spoiler for

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle

firstintersects a trisector of one of the other angles.And there is something special about those three points.

Spoiler for i think

Actually they are not collinear.

That being the case, they form a triangle.

So the OP really asks: what is special about the triangle formed by these three points?

*Vidi vici veni.*

### #6

Posted 14 March 2013 - 09:47 PM Best Answer

### #7

Posted 15 March 2013 - 05:25 AM

Spoiler for It seems...

Correct. Good job.

I wonder if there is a proof of this that is not overly complex?

Edit:

Well, No. I just found the proof, and it's not beautiful for its simplicity.

You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.

"Do not try this at home."

**Edited by bonanova, 15 March 2013 - 06:23 AM.**

Comment on proof

*Vidi vici veni.*

### #8

Posted 18 March 2013 - 01:01 PM

Spoiler for It seems...Correct. Good job.

I wonder if there is a proof of this that is not overly complex?

Edit:

Well, No. I just found the proof, and it's not beautiful for its simplicity.

You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.

"Do not try this at home."

oh i am so sorry .....i mistook it for the eulers line....my bad...

### #9

Posted 24 March 2013 - 03:09 AM

COMBINATION: 7129

#### 0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users